This research project is concerned with elastic waves, and the main purposes set initially werea) to obtain concrete representations of the solutions,b) to study scattering of the waves near boundaries,c) to study inverse problems concerning the waves near boundaries,d) to analyze the waves in the case of the total reflection.We have accomplished these almost as was expected. Let us summarize the results obtained in this project.About a) and d) , we have got an asymptotic expansion of the wave reflected totally, which is one of the mainest results. This expansion is much expected to be useful in analyzing the phenomenon of the scattering and the inverse problems of the elastic waves. Furthermore, we have obtained also another kind of concrete representation of the waves.About b) , we have shown that the Rayleigh wave, which has been much interested since a long time ago, behaves individually and can be extracted in the Lax-Phillips scattering theory : We have made a formulation of that theory for the Rayleigh wave, which is useful for the inverse problems. Moreover, we have investigated precisely decay of this wave expressing the behavior at infinity.About c) , we have got a new methods of reconstruction of coefficients in differential equations applicable to the inverse problems. And we have obtained a new numerical method of finite elements for the approximate solutions. These seem to be very useful to solve the inverse problems, but we cannot finish solving concretely some of the inverse problems.